Multiway System
A multiway system is a kind of substitution system in which multiple states are permitted at any stage. This accommodates rule systems in which there is more than one possible way to perform an update.
A simple example is a string substitution system. For instance, take the rules 👁 {AB->A,BA->B}
and the initial condition 👁 ABA
. There are two choices for how to proceed. Applying the
first rule yields the evolution 👁 ABA toAA
, while applying the second rule yields the evolution
👁 ABA->AB->A
. So at the first step,
there is a single state (👁 {ABA}
), at the second step there are two states 👁 {AA,AB}
, and at the third step there is a single state 👁 {A}
.
A path through a multiway system arising from a choice of which substitutions to make is called an evolution. Typically, a multiway system will have a large number
of possible evolutions. For example, consider strings of 👁 A
s and 👁 B
s with the rule 👁 AB->BA
. Then most strings will have more than one occurrence
of the substring 👁 AB
,
and each occurrence leads down another path in the multiway system.
See also
Branchial Graph, Branchial Space, Causal Invariance, Confluent, Multicomputational Paradigm, Multiway Graph, Reduction System, Sequential Substitution System, Substitution System, Term Rewriting SystemThis entry contributed by Todd Rowland
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References
Gorard, J.; Wolfram, S.; and Piskunov, M. "MultiwaySystem." https://resources.wolframcloud.com/FunctionRepository/resources/MultiwaySystem.Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, pp. 204-209 and 937-939, 2002.Wolfram, S. "Multicomputation with Numbers: The Case of Simple Multiway Systems." 9 Nov 2021. https://arxiv.org/abs/2111.04895.Wolfram, S. "Expression Evaluation and Fundamental Physics." Sep. 29, 2023. https://writings.stephenwolfram.com/2023/09/expression-evaluation-and-fundamental-physics/.Referenced on Wolfram|Alpha
Multiway SystemCite this as:
Rowland, Todd. "Multiway System." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/MultiwaySystem.html